Theory of the Two-Impurity Kondo Effect in the Presence of an Impurity-Impurity Exchange Interaction

We examine the two-impurity Kondo effect by deriving the equation of motion of a set of Green's functions using the two-impurity s-d Hamiltonian with an added exchange term of the form W(→S₀·→S₁), where →S₀ and →S₁ are the spin operators of the two impurities. The resulting Green's functions are truncated and solved for self-consistency, keeping the most divergent terms. Our results show that all the lnT terms arising in the single-impurity Kondo effect are modified and replaced by ln(T²+W^'²)1/2, where W^' is an energy approximately equal to W. This results in an effective Kondo temperature TKE, where TKE=TK⁰[1-(W^'TK⁰)²]1/2, and TK⁰ is the single-impurity Kondo temperature. Thus the effective Kondo temperature decreases as the impurity-impurity interaction increases, and when W^' is greater than TK⁰ the Kondo divergence is removed by the impurity-impurity interaction. Our results show that the interaction W strongly modifies the spin-compensated state. We also derive expressions for the conduction-electron polarization as a function of W^' for high values of W^' or high temperatures.